Figurate Numbers and Sums of Numerical Powers: Fermat, Pascal, Bernoulli
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Figurate numbers and sums of numerical powers: Fermat, Pascal, Bernoulli David Pengelley In the year 1636, one of the greatest mathematicians of the early seventeenth century, the Frenchman Pierre de Fermat (1601—1665), wrote to his correspondent Gilles Personne de Roberval1 (1602—1675) that he could solve “what is perhaps the most beautiful problem of all arithmetic” [4], and stated that he could do this using the following theorem on the figurate numbers derived from natural progressions, as he first stated in a letter to another correspondent, Marin Mersenne2 (1588—1648) in Paris: 11111111 Pierre de Fermat, from Letter to Mersenne. September/October, 1636, and again to Roberval, November 4, 1636 The last number multiplied by the next larger number is double the collateral triangle; the last number multiplied by the triangle of the next larger is three times the collateral pyramid; the last number multiplied by the pyramid of the next larger is four times the collateral triangulo-triangle; and so on indefinitely in this same manner [4], [15, p. 230f], [8, vol. II, pp. 70, 84—85]. 11111111 Fermat, most famous for his last theorem,3 worked on many problems, some of which had ancient origins. Fermat had a law degree and spent most of his life as a government offi cial in Toulouse. There are many indications that he did mathematics partly as a diversion from his professional duties, solely for personal gratification. That was not unusual in his day, since a mathematical profession comparable to today’sdid not exist. Very few scholars in Europe made a living through their research accomplishments. Fermat had one especially unusual trait: characteristically he did not divulge proofs for the discoveries he wrote of to others; rather, he challenged them to find proofs Mathematical Sciences; Dept. 3MB, Box 30001; New Mexico State University; Las Cruces, NM 88003; [email protected]. 1 Roberval was a prominent mathematician of the era just before the invention of the infinitesimal calculus. He developed his own “method of indivisibles”for finding areas and volumes before the fundamental theorem of calculus was available. For more on his life and work see, e.g., [19]. 2 Mersenne was a theologian, philosopher, mathematician and music theorist. In addition to his creative work, he carried on extensive correspondence with mathematicians and other scientists in many countries. The scientific journal had not yet come into being, and Mersenne served as the central node for exchange of much mathematical and scientific information. For more on his life and work see, e.g., [16]. 3 Fermat claimed that the equation xn +yn = zn has no solution in positive whole numbers x, y, z when n > 2. One of the greatest triumphs of twentieth-century mathematics was the proof in the 1990s of his famous long-standing claim. This was accomplished by Andrew Wiles, utilizing much of the advanced number theory developed in the preceding century. For more information see, e.g., [9]. 1 Figure 1: Fermat of their own. While he enjoyed the attention and esteem he received from his correspondents, he never showed interest in publishing a book with his results. He never traveled to the centers of mathematical activity, not even Paris, preferring to communicate with the scientific community through an exchange of letters, facilitated by the Parisian theologian Marin Mersenne, who served as a clearinghouse for scientific correspondence from all over Europe, in the absence as yet of scientific research journals. While Fermat made very important contributions to the development of the differential and integral calculus and to analytic geometry [15, Chapters III, IV], his life-long passion belonged to the study of properties of the integers, now known as number theory, and it is here that Fermat has had the most lasting influence on the subsequent course of mathematics. In hindsight, Fermat was one of the great mathematical pioneers, who built a whole new paradigm for number theory on the accomplishments of his predecessors, and laid the foundations for a mathematical theory that would later be referred to as the “queen of mathematics.”[17] What was this “most beautiful problem of all arithmetic” to which Fermat refers in his letter to Roberval? And what are the figurate numbers about which he makes his geometric sounding claims? He writes as if these numbers are themselves geometric figures like triangles and pyramids. In fact they are the numbers that “count”how many equally spaced dots occur in forming discrete versions of these geometric figures, and such figurate numbers had first been studied millennia earlier. Fermat’s“most beautiful problem of all arithmetic”, which he claimed he could solve using figurate numbers, also had a long history. He was referring to finding formulas for sums of powers in a natural progression, i.e., in an arithmetic progression 1, 2, . , n whose “last number” is n. We shall see shortly what is meant by a sum of powers, and see why formulas for sums of powers became central in mathematics as the ideas of calculus emerged through the seventeenth century. Both figurate numbers and formulas for sums of powers were important in mathematics long before and after Fermat, and the whole story is told in full in [14]. This project focuses on the 2 intertwined study of the figurate numbers and formulas for sums of powers through the work of three great seventeenth century mathematicians, Pierre de Fermat, Blaise Pascal (1623—1662),and Jakob Bernoulli (1654—1705). We will see that while figurate numbers are not generally well known by that name today, they are the same numbers that count combination of choices, that appear as coeffi cients in binomial expansions, and that appear in Pascal’sTriangle, and are thus still perhaps the most important numbers in combinatorial mathematics today. Fermat’sfigurate numbers and sums of powers Already in the classical Greek era, figurate numbers were of great interest, and formulas for sums of powers were sought for solving area and volume problems. The Pythagoreans, a mysterious group led by Pythagoras in ancient Greece around the sixth century b.c.e., believed that number was the substance of all things, and geometric patterns seen in dots or pebbles in the sand showed them relationships between numbers. For instance, in Figure 2 we see dot pictures for three types of numbers the Pythagoreans probably considered, and we can obtain some formulas for discrete sums from them. Figure 2: Square, rectangular, and triangular numbers Exercise 1. Using the dot pictures of Figure 2, explain how to obtain summation formulas like n n n n (n + 1) (2i 1) = n2, 2i = n (n + 1) , i = , 2 i=1 i=1 i=1 X X X and then prove these formulas by mathematical induction. We can now also understand Fermat’sfirst claim, if we realize that for the progression 1, 2, . , n, what Fermat means by the collateral triangle is the triangle with rows of dots whose quantity ranges from 1 through n, as in Figure 2. We will call the total number of dots in such a triangle a triangular number. Exercise 2. Interpret and justify Fermat’sfirst claim about the collateral triangle using the results of the previous exercise. The further figurate numbers, such as Fermat’scollateral pyramid, will generalize this by count- ing dots in analogous higher-dimensional figures. We shall study these shortly, but first we begin the parallel story of sums of powers. In classical Greek mathematics we find an interest not just in knowing how to sum the terms in 4 n n(n+1) a progression, as in the closed formula appearing on the right side of i=1 i = 2 . We also find n 2 n 3 sums like i=1 i and i=1 i , which today we call sums of powers, i.e., the sum of n terms in the progression each raised to a fixed power. These arose in specific determinationsP of areas and volumes P P 4 By a closed formula we mean one without an arbitrarily long summation process. 3 of geometric objects with curved sides, by what is today called the Greek method of exhaustion. For instance, Archimedes of Syracuse (c. 287—212 b.c.e.), perhaps the greatest mathematician of antiquity, determined areas bounded by parabolas and spirals. To see the connection to sums of powers, the reader should recall and review from modern calculus that an area bounded by curves is given by an integral, defined as a limit of sums that approximate the area with sums of areas 1 k of rectangles. Recall, for instance, that for the important integral 0 x dx, with k an arbitrary fixed natural number, the simplest Riemann sums will involve n ik, and that to determine the i=1 R integral as a limiting value of the Riemann sums, one will need some very good knowledge about the size of these sums as n grows. While you may be temptedP simply to evaluate this integral by antidifferentiation using the fundamental theorem of calculus, note that neither antidifferentiation nor the fundamental theorem of calculus were even dreamt of by anyone in the world until the second half of the seventeenth century, two millennia after Archimedes and several decades after Fermat’s work. However, by Fermat’s time in the first half of the seventeenth century, finding formulas for n k 1 k sums of powers i=1 i for the purpose of computing the integrals 0 x dx was a major quest in mathematics. We should therefore not be surprised that Archimedes desired knowledge about sums n P2 1 2 R of squares i=1 i , since 0 x dx gives an area bounded by a parabola, and the reader may check that the same integral arises when finding the area bounded by an Archimedean spiral, a curve R described todayP by r = a in polar coordinates.